Let me make the initial velocity parameter clear. It is being
expressed in terms of the velocity for circular orbit Vo. You
should be able to get a circular orbit by choosing Vo as the
velocity of projection and 90 degrees for angle of projection.
This would be

This value is about 8 km/s for close earth
orbits. ( Use R, the radius of the earth as the distance and an
angle of projection of 90 degrees to see this). Note that The
magnitude of velocity during subsequent motion changes if it is
not a circular orbit.

You should be able to see the independence of time period on
eccentricity by changing the angle of projection (Letting other
parameters remain the same). It should be remembered that the
major axis depends only on the energy and time period only the on
the major axis. Since you are choosing a certain value for the
distance from the earth for projection and are projecting at the
same velocity at different angles the energy is same and hence
the major axis. I am putting down the relevant relations here for
your reference

and this in our case is equal to

The energy at the intial moment is dependent on r and v only
and is unchanged by changing the angle of projection. The time
period for a given orbit is

Try experimenting with cirucular orbits of radii
in the ratio 1:4. You should be able to relate to the result of
the relation above.

You could also see interesting situations of
angular momentum conservation. Try choosing an angle of
projection of 60 degrees when radius vector is 2*R and the
initial velocity Vo. You would see the satellite grazing past the
surface of the earth in its path. (Can you see why?) You should
be able to see other possibilities.